The context:
for my research I am currently looking at parabolic systems of the type
$$
\left\{
\begin{array}{ll}
\partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\
u=0 & x\in\Gamma=\partial\Omega\\
u(t=0)=u_0
\end{array}
\right.\hspace{2cm}(0)
$$
where $\Omega\subset \mathbb{R}^d$ is a smooth bounded domain and $u=(u_1,\ldots,u_k)$ is vector valued. The vector field $b:\mathbb{R}^k\to\mathbb{R}^k$ is monotone in the sense that
$$
\forall z_1,z_2\in \mathbb{R}^k:\qquad (b(z_1)-b(z_2))\cdot(z_1-z_2)\geq 0\quad\text{and}\quad b(0)=0\hspace{3cm}(1)
$$
so the problem has a nice variational structure. The main (and multiply cited) reference I found for this type of problems is [Alt-Luckhaus, Quasilinear elliptic-parabolic differential equations, '83]. They actually consider a much more general problem with non-homogenous Dirichlet boundary values, nonlinear diffusion $\Delta u\leftrightarrow div(a(b(u),\nabla u))$ and additional reaction term $f(b(u))$, but let me keep things really simple here. I am mostly interested in the degenerate case, a typical example is $b(u)=u|u|^{p-1}$ for fixed $p\in(0,1)$. Degeneracy means here that the derivative $Db(u)$ blows-up near $u=0$ (in the example $Db(u)=\mathcal{O}(u^{p-1})\to\infty$), even though this is not important for my purpose here.

Convex variational structure and Legendre transform:
In the case when $b$ is the gradient of a $\mathcal{C}^1$ convex function
$$
b=\nabla\Phi,\quad \Phi\text{ convex and wlog }\Phi(0)=\min\Phi=0,
$$
we have of course monotonicity in the sense of (1). But more importantly, in this case the variational structure of the problem is automatically convex. More precisely, define the usual Legendre transform
$$
\Psi(p)=\Phi^*(p)\overset{\text{def}}{=}\sup\limits_{z\in \mathbb{R}^k}\,(p\cdot z-\Phi(z))
$$
and set
$$
B(z):=\Psi(\nabla\Phi(z))=\Psi(b(z)).
$$
Since $\Phi$ is convex we have by usual properties that $B(z)=\nabla\Phi(z)\cdot z-\Phi(z)=b(z)\cdot z-\Phi(z)$ for all $z$ (if $p=\nabla\Phi(z_0)$ for some $z_0$ then the sup in the definition of $\Phi^*$ is attained preciesly for $z=z_0$ since $\Phi$ is convex). Hence for smooth solutions
$$
\partial_tb(u)\cdot u=\partial_t(B(u)).
$$
Formally testing $\zeta=u$ as a test function in (0) we get the a priori estimate
$$
\forall T>0:\qquad\sup\limits_{t\in[0,T]}\int_{\Omega}B(u(t))+\int_0^T|\nabla u(t)|^2_{L^2(\Omega)}\leq \int_{\Omega}B(u_0)\hspace{4cm}(2).
$$
This is particularly enlightening for the linear heat equation $b(z)=z=D(|z^2|/2)$ and $ B(z)\propto |z|^2$, in which case (2) is nothing but the usual energy estimate $\qquad\sup\limits_{t\in[0,T]}|u(t)|_{L^2(\Omega)}+\int_0^T|\nabla u(t)|^2_{L^2(\Omega)}\leq |u_0|_{L^2(\Omega)}$ (up to multiplicative constants).

My question(s): what to do when there is no intrinsic convexity, i-e when $b$ is monotone but not the gradient of a ($\mathcal{C}^1$) convex function $\Phi$? The problem is still variational so something could still be done... Let me elaborate a litte:

Do you know examples of monotone vector fields in the sense of (1) that are not convex gradients? I guess any subgradient $b(z)\in\partial\Phi(z)$ would do, but I am not familiar with non-smooth convex analysis.

in the convex case $b=\nabla\Phi$ the "infinitesimal equation" $\partial b(u)\cdot u=\partial_t(B(u))$ holds at least for smooth solutions, and leads to the a priori estimate (2). But it turns out that the convexity is completely encoded in the weaker "monotonicity/convexity" inequality
$$
\forall z_1,z_2\in \mathbb{R}^k:\qquad (b(z_1)-b(z_2))\cdot z_1\geq B(z_1)-B(z_2)\hspace{2cm}(3).
$$
This holds of course when $b=\nabla\Phi$ with the above definition of $B$. It is easy to check that, whenever $b,B$ satisfy (3), one can get the energy estimate (2) (I'll skip the details here). In the convex case this real-valued function $B(z)$ is constructed as above by naturally using the "intrinsic" convexity and Legendre transform. In the non-convex case, is there a natural way to construct a suitable (real-valued) function $B(z)$ such that (3) holds? By suitable I mean that $B$ should be at least nonnegative, but also preferably convex in the solution $u$ (for the heat equation $B(u)\propto |u|^2$).

My ideas:

In the smooth convex case the classical computation ensuring that $b=\nabla\Phi$ is monotone (1) is: if $z_s=sz_1+(1-s) z_2$ is the segment $[z_2,z_1]$ then
\begin{align*}
(b(z_1)-b(z_2))\cdot(z_1-z_2)& =\int_0^1\frac{d}{ds}\left(b(z_s)\cdot(z_1-z_2)\right)ds\\
& =\int_0^1\underbrace{Db(z_s)}_{=D^2\Phi(z_s)\geq 0}(z_1-z_2)\cdot(z_1-z_2)ds\geq 0.
\end{align*}
When $b$ is not a gradient the first equalities still hold so we see that, in order to ensure that $b$ is monotone, it is enough to suppose that the symmetric part of its gradient
$$
S(z):=\left[\frac{Db+(Db)^t}{2}\right](z)\geq 0
$$
is positive semi-definite for all $z\in \mathbb{R}^k$ (the antisymmetric part $A=Db-S$ gives of course a zero contribution $A(z_s)(z_1-z_2)\cdot(z_1-z_2)=0$ for all $s$). I was hoping to construct the function $B$ using the symmetric part of its gradient, anyone knows a way to do that?

Is a monotone vector field $b$ in the sense of (1) automatically the subgradient $b(z)\in\partial \Phi(z)$ of a convex function? This vaguely reminds me of something in the book [Rockafellar, convex analysis], so I guess there is a well known answer and I apologize for asking. If so, there may be a natural way to obtain a convexity/monotonicity inequality in the spirit of (3), where I guess $B$ should be constructed using again the Legendre transform $\Phi^*(p)$.

Thank you for your time and comments, any input will be much appreciated!

1 Answer
1

Let $\lambda>0$ and $A$ be any non-symmetric $d$-by-$d$ matrix such that $Ax\cdot x \geq \lambda |x|^2$ for every $x\in\mathbb{R}^d$. Then $x\mapsto Ax$ is uniformly monotone but is not the gradient of a convex function.

What you are looking for (duality/variational theory for monotone maps, and what replaces the Legendre transform, etc) is not-very well known, but it is known. I suggest to look in Section 2 of this paper and the references there: http://arxiv.org/abs/1411.3668